SUSY, Spin Chains (andASEPS?)

Des JohnstonMaxwell Institute for Mathematical Sciences

Heriot-Watt University, EdinburghCPPT, Sept 2017

Johnston SUSY 1/29

Nerses

Congratulations (and thanks)!

Symbols of Armenia

Johnston SUSY 2/29

Plan of talk

SUSY

(XXZ) Spin Chains/SUSY - Fendley, Hagendorf

The missing punchline - ASEPs (ASAPs) - Ayyer, Mallick

NOT Johnston

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SUSYOriginally.....

Q|Boson〉 = |Fermion〉

Q†|Fermion〉 = |Boson〉

Q2 = 0,(

Q†)2

= 0

In particular, Super-Poincaré algebra in 4d

{Qα, Q̄β̇} = 2(σµ)αβ̇Pµ

Getting rid of divergences in QFT

Lattice discretisation difficult (but see Catterall et.al.)

Johnston SUSY 4/29

SUSY QMHamiltonian

Hψn(x ) =

(−~2

2md 2

dx 2 +mω2

2x 2)ψn(x ) = Enψn(x )

Define

A =~√2m

ddx

+ W (x ); A† = − ~√2m

ddx

+ W (x )

Two SUSY Hamiltonians

H (1) = A†A =−~2

2md 2

dx 2 −~√2m

W ′(x ) + W 2(x )

H (2) = AA† =−~2

2md 2

dx 2 +~√2m

W ′(x ) + W 2(x )

Johnston SUSY 5/29

SUSY QM IISUSY algebra

[H ,Q] = [H ,Q†] = 0{Q,Q†} = H{Q,Q} = {Q†,Q†} = 0

H =

(H (1) 0

0 H (2)

)Q =

(0 0A 0

)Q† =

(0 A†

0 0

)

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XXZ Spin Chain

XXZ Hamiltonian acts on V⊗N where V ' C2.

H (N) = −12

N−1∑i=1

(σx

i σxi+1 + σy

i σyi+1 −

12σz

i σzi+1

)−1

4(σz

1 + σzN ) +

3N − 14

I

Write as sum of local terms, hij : V ⊗ V → V ⊗ V in bulk

H (N) =∑〈ij〉

hij + h1 + hN

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XXZ Spin Chain II

Local Hamiltonian (in the bulk)

hij = σx ⊗ σx + σy ⊗ σy − 12σz ⊗ σz +

34I

In matrix form, standard basis for C2 ⊗ C2

hij =

1 0 0 00 1

2 −1 00 −1 1

2 00 0 0 1

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XXZ Spin Chain/SUSY

Possible to define (Fendley, Yang)

Q(N) : V⊗N → V⊗N−1; Q(N)† : V (N−1) → V (N)

Properties of Q: nilpotent

Q(N−1)Q(N) = 0, Q†(N+1)Q†(N) = 0,

Hamiltonian length N

H (N) = Q†(N)Q(N) + Q(N+1)Q(N+1)†.

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XXZ Spin Chain/Supercharge I

Length-changing operator

H (N−1)Q(N) = Q(N)H (N), H (N)Q†(N) = Q†(N)H (N−1).

Local expressions (Hagendorf)

Q(N) =N−1∑i=1

(−1)i+1qi ,i+1, Q†(N) =N−1∑

i

(−1)i+1q†i

Local action - “dynamical SUSY”

q†i : V → V ⊗ V ; qi ,i+1 : V ⊗ V → V

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XXZ Spin Chain/Supercharge II

In terms of the local supercharge, nilpotency (Q†)2 = 0becomes

(q† ⊗ 1− 1⊗ q†)q† = 0

Or, more accurately - including a sort of “gauge” freedom[(q† ⊗ 1)q† − (1⊗ q†)q†

]|m〉 = |χ〉 ⊗ |m〉 − |m〉 ⊗ |χ〉

For all |m〉 ∈ V , for a particular |χ〉 ∈ V → V ⊗ V

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XXZ Spin Chain/Supercharge III

Substitute Q,Q† expression into Hamiltonian, most termsvanish:

q†i qj + qj+1q†i = 0, 1 ≤ i < j − 1 ≤ N − 1

Leaving

h = −(q ⊗ 1)(1⊗ q†)− (1⊗ q)(q† ⊗ 1) + q†q

+12

(q q† ⊗ 1 + 1⊗ q q†

)With boundary terms

hB =12

q q†

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XXZ Spin Chain/Supercharge IIIbis

The game of “does my Hamiltonian have dynamicalsupersymmetry”?

1) Pick a q† - (4× 2)

2) Construct h - (4× 4) using:

h = −(q ⊗ 1)(1⊗ q†)− (1⊗ q)(q† ⊗ 1) + q†q

+12

(q q† ⊗ 1 + 1⊗ q q†

)

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XXZ Spin Chain/Supercharge IV

q† is remarkably simple for XYZ (at combinatorial point)

q† =

0 10 00 00 0

hij =

1 0 0 00 1

2 −1 00 −1 1

2 00 0 0 1

With

|0〉 =

(10

); |1〉 =

(01

)Action

q†|0〉 = ∅ ; q†|1〉 = |00〉

Johnston SUSY 14/29

XXZ Spin Chain/Supercharge VCould start with |1〉 instead of |0〉, i.e.

q̄†|1〉 = ∅ ; q̄†|0〉 = |11〉

q̄† is still simple

q̄† =

0 00 00 01 0

hij =

1 0 0 00 1

2 −1 00 −1 1

2 00 0 0 1

Stitch together q, q̄, using “gauge” freedom

q†(y ) = x

−2y 1−y 2 −y−y 2 −yy 3 −2y 2

; x = (1 + |y |6)−1/2

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XXZ Spin Chain/Supercharge VIWith q†(y ) still true that

hij =

1 0 0 00 1

2 −1 00 −1 1

2 00 0 0 1

Boundaries become (y = ρe iθ)

hB(y ) =

(1 + 5ρ2 + ρ4

4(1− ρ2 + ρ4)

)I +

3∑j=1

λjσj ,

where

λ1 = −ρ cos θ

1 + ρ2 , λ2 = − ρ sin θ

1 + ρ2 , λ3 = −14

(1− ρ2

1 + ρ2

).

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XXZ Spin Chain/Supercharge VII

So far - SUSY for XXZ with off-diagonal BCs same at bothsends

hij = σx ⊗ σx + σy ⊗ σy − 12σz ⊗ σz +

34I

hB(y ) =

(1 + 5ρ2 + ρ4

4(1− ρ2 + ρ4)

)I +

3∑j=1

λjσj ,

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XXZ Spin Chain/Supercharge VIII

Can also arrange different BCs, take

|φ(y )〉 =1∑

m=0

φm(y )|m〉, φm(y ) = − y m+1√{m + 1}

.

|ξk (y )〉 = x (|φ(y )〉 − |φ(q2(k+1)y )〉), q = − exp(−iπ/3)

Boundary vectors

q(y )|ξk (y )〉 = |ξk (y )〉 ⊗ |ξk (y )〉, k = 0,1,2

Then

Qj ,k (y )|ψ〉 = |ξj (y )〉⊗|ψ〉+(−1)N−1|ψ〉⊗|ξk (y )〉+Q(y )|ψ〉.

Johnston SUSY 18/29

XXZ Spin Chain/Supercharge IX

Now - SUSY for XXZ with off-diagonal BCs different atboths ends

hij = σx ⊗ σx + σy ⊗ σy − 12σz ⊗ σz +

34I

hB(y ) =

(1 + 5ρ2 + ρ4

4(1− ρ2 + ρ4)

)I +

3∑j=1

λjσj ,

h(k )B (y ) = hB(q2(k+1)y )

q = − exp(−iπ/3)

Johnston SUSY 19/29

Random additional Comments

XYZ too

hij = (1− z)σx ⊗ σx + (1 + z)σy ⊗ σy − 1− z2

2σz ⊗ σz

q† in this case

q† =

0 10 00 00 −z

Hard fermion models: M2, Ml - Hagendorf, Huijse

Gl(n|m) - Meidinger, Mitev

Johnston SUSY 20/29

Story so Far

XXZ (and XYZ) spin chain has a “dynamical”supersymmetry

Relates chains of different length

Can be formulated for both open and closed chains

Can be formulated for diagonal, off-diagonal, unequal BCs

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ASEPs

Originally came from biology

Model for transport in cells

Kinesins moving along a microtubule

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The Phase Diagram (q = 0)

21

21 α

β

LD MC

HD

Roll of honour (TASEP q = 0): Derrida,Evans, Hakim PasquierRoll of honour (PASEP q 6= 0): Blythe, Evans, Colaiori, Essler

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What’s this got to do with XXZ ?Markov matrix M

ddt|P(t )〉 = M |P(t )〉,

Related to HXXZ

M =√

pqU−1µ HXXZ Uµ

Where

Uµ =L⊗

i=1

(1 00 µQ j−1

),

HXXZ = −12

L−1∑j=1

[σx

j σxj+1 + σ

yj σ

yj+1 −∆σz

j σzj+1

+h(σzj+1 − σ

zj ) + ∆

]+ B1 + BL

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What’s this got to do with XXZ ??

Various XXZ parameters related to (P)ASEP

B1 =1

2√

pq

(α + γ + (α− γ)σz

1 − 2αµeλσ−1 − 2γµ−1e−λσ+1)

BL =1

2√

pq

(β + δ − (β − δ)σz

L − 2δµQL−1σ−L

− 2βµ−1Q−L+1σ+L

)∆ = −1

2(Q + Q−1), h =

12

(Q −Q−1), Q =

√qp.

Johnston SUSY 25/29

And SUSY??Well.....“transfer matrix” symmetry (Ayyer, Mallick)

For ASEP and ASAP

TL,L+1ML = ML+1TL,L+1

Relates Ms of different length

ML =L−1∑i=1

Mi ,i+1 + R + L

Where, for ASAP

10→ 01 with rate 111→ 00 with rate λ

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And SUSY???

ASAP local Hamiltonian

hi ,i+1 = σ+i σ−i+1 + λσ+i σ

+i+1 +

1− λ4

σzi σ

zi+1 +

1 + λ

4σz

i

− 1− λ4

σzi+1 −

1 + λ

4I

L = α

(σ−1 + λσ+1 −

1− λ2

σz1 −

1 + λ

2

)R = β

(σ+L −

1− σzL

2

)

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So??

Is the transfer matrix symmetry related to SUSY?

Related spin chains have unequal, off-diagonal BCs

NOT at combinatorial point <<<<

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References

Xiao Yang and Paul Fendley, Non-local spacetimesupersymmetry on the lattice, J. Phys. A,37(38):8937–8948, 2004.

Christian Hagendorf and Jean Liénardy, Open spin chainswith dynamic lattice supersymmetry, J. Phys. A,50(18):185202, 32, 2017.

Robert Weston and Junye Yang, Lattice Supersymmetry inthe Open XXZ Model: An Algebraic Bethe Ansatz Analysis,[arXiv:1709.00442]

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